Activation Functions as Mathematical Operators
After a neural network computes a linear transformation of its input — such as multiplying by a weight matrix and adding a bias — it applies a function called an activation function to each component of the result. Activation functions are applied pointwise: for each output of the linear map, you independently transform it using the same mathematical rule. This operation introduces nonlinearity into the network, which is crucial for modeling complex, real-world relationships that cannot be captured by linear functions alone.
An activation function is a mathematical function applied to each element of a vector (or matrix) output by a linear transformation in a neural network. Common examples include:
- ReLU (Rectified Linear Unit): f(x)=max(0,x);
- Sigmoid: f(x)=1/(1+exp(−x));
- Tanh: f(x)=(exp(x)−exp(−x))/(exp(x)+exp(−x)).
Each of these functions transforms its input in a specific way, introducing nonlinearity and controlling the range of possible outputs.
If you only stack linear transformations (matrix multiplications and additions), the result is always another linear transformation. No matter how many layers you add, the network can only model linear relationships, which are far too simple for most real-world tasks. Nonlinearity allows the network to "bend" and "reshape" data in ways that capture complex patterns.
Activation functions break the strict linearity of the network's computations. By applying a nonlinear function after each linear map, you enable the network to approximate any continuous function on a compact domain — a property known as universal approximation. This is only possible because the activation function disrupts the direct proportionality of input and output, making the network fundamentally more expressive.
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After a neural network computes a linear transformation of its input — such as multiplying by a weight matrix and adding a bias — it applies a function called an activation function to each component of the result. Activation functions are applied pointwise: for each output of the linear map, you independently transform it using the same mathematical rule. This operation introduces nonlinearity into the network, which is crucial for modeling complex, real-world relationships that cannot be captured by linear functions alone.
An activation function is a mathematical function applied to each element of a vector (or matrix) output by a linear transformation in a neural network. Common examples include:
- ReLU (Rectified Linear Unit): f(x)=max(0,x);
- Sigmoid: f(x)=1/(1+exp(−x));
- Tanh: f(x)=(exp(x)−exp(−x))/(exp(x)+exp(−x)).
Each of these functions transforms its input in a specific way, introducing nonlinearity and controlling the range of possible outputs.
If you only stack linear transformations (matrix multiplications and additions), the result is always another linear transformation. No matter how many layers you add, the network can only model linear relationships, which are far too simple for most real-world tasks. Nonlinearity allows the network to "bend" and "reshape" data in ways that capture complex patterns.
Activation functions break the strict linearity of the network's computations. By applying a nonlinear function after each linear map, you enable the network to approximate any continuous function on a compact domain — a property known as universal approximation. This is only possible because the activation function disrupts the direct proportionality of input and output, making the network fundamentally more expressive.
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